Determine how many solutions exist for the system of equations. ${-4x+y = -6}$ ${y = 1+3x}$
Solution: Convert both equations to slope-intercept form: ${-4x+y = -6}$ $-4x{+4x} + y = -6{+4x}$ $y = -6+4x$ ${y = 4x-6}$ ${y = 1+3x}$ ${y = 3x+1}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 4x-6}$ ${y = 3x+1}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.